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Mirrors > Home > HSE Home > Th. List > bdophsi | Structured version Visualization version GIF version |
Description: The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoptri.1 | ⊢ 𝑆 ∈ BndLinOp |
nmoptri.2 | ⊢ 𝑇 ∈ BndLinOp |
Ref | Expression |
---|---|
bdophsi | ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | . . . 4 ⊢ 𝑆 ∈ BndLinOp | |
2 | bdopln 31748 | . . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆 ∈ LinOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝑆 ∈ LinOp |
4 | nmoptri.2 | . . . 4 ⊢ 𝑇 ∈ BndLinOp | |
5 | bdopln 31748 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 𝑇 ∈ LinOp |
7 | 3, 6 | lnophsi 31888 | . 2 ⊢ (𝑆 +op 𝑇) ∈ LinOp |
8 | bdopf 31749 | . . . . . 6 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ 𝑆: ℋ⟶ ℋ |
10 | bdopf 31749 | . . . . . 6 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
11 | 4, 10 | ax-mp 5 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ |
12 | 9, 11 | hoaddcli 31655 | . . . 4 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
13 | nmopxr 31753 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → (normop‘(𝑆 +op 𝑇)) ∈ ℝ*) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ* |
15 | nmopre 31757 | . . . . 5 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ) | |
16 | 1, 15 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑆) ∈ ℝ |
17 | nmopre 31757 | . . . . 5 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
18 | 4, 17 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑇) ∈ ℝ |
19 | 16, 18 | readdcli 11266 | . . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ |
20 | nmopgtmnf 31755 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝑆 +op 𝑇))) | |
21 | 12, 20 | ax-mp 5 | . . 3 ⊢ -∞ < (normop‘(𝑆 +op 𝑇)) |
22 | 1, 4 | nmoptrii 31981 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) |
23 | xrre 13188 | . . 3 ⊢ ((((normop‘(𝑆 +op 𝑇)) ∈ ℝ* ∧ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ) ∧ (-∞ < (normop‘(𝑆 +op 𝑇)) ∧ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)))) → (normop‘(𝑆 +op 𝑇)) ∈ ℝ) | |
24 | 14, 19, 21, 22, 23 | mp4an 691 | . 2 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ |
25 | elbdop2 31758 | . 2 ⊢ ((𝑆 +op 𝑇) ∈ BndLinOp ↔ ((𝑆 +op 𝑇) ∈ LinOp ∧ (normop‘(𝑆 +op 𝑇)) ∈ ℝ)) | |
26 | 7, 24, 25 | mpbir2an 709 | 1 ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 class class class wbr 5149 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℝcr 11144 + caddc 11148 -∞cmnf 11283 ℝ*cxr 11284 < clt 11285 ≤ cle 11286 ℋchba 30806 +op chos 30825 normopcnop 30832 LinOpclo 30834 BndLinOpcbo 30835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-hilex 30886 ax-hfvadd 30887 ax-hvcom 30888 ax-hvass 30889 ax-hv0cl 30890 ax-hvaddid 30891 ax-hfvmul 30892 ax-hvmulid 30893 ax-hvmulass 30894 ax-hvdistr1 30895 ax-hvdistr2 30896 ax-hvmul0 30897 ax-hfi 30966 ax-his1 30969 ax-his2 30970 ax-his3 30971 ax-his4 30972 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9472 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14008 df-exp 14068 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 df-grpo 30380 df-gid 30381 df-ablo 30432 df-vc 30446 df-nv 30479 df-va 30482 df-ba 30483 df-sm 30484 df-0v 30485 df-nmcv 30487 df-hnorm 30855 df-hba 30856 df-hvsub 30858 df-hosum 31617 df-nmop 31726 df-lnop 31728 df-bdop 31729 |
This theorem is referenced by: bdophdi 31984 nmoptri2i 31986 unierri 31991 |
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